
% \section{Overview and Related works}
\section{Introduction}

The introduction of smart phones, such as the iPhone and Android, has
resulted in a strong increase in mobile data traffic. According to the
report\cite{cisco_data}, on average, a smart phone generates 24 times
more data traffic than a feature phone. Due to this increase in data
traffic, mobile service providers are facing system overload. Among
the traffic explosion, data traffic from indoor area (e.g., home,
office, etc.) is major concern for service providers, because the
channel quality between macro BSs and indoor areas are suffered from
significant pathloss due to walls and users spend about 60\% of times
in home or office. One way to address this problem is for the
providers to increase their system capacity. However, upgrading
systems is itself problematic: it is very expensive and the amount of
data traffic is increasing at a rate that outstrips the ability to
upgrade the systems that handle it.

An alternative solution to the problem of data overload is to offload
data to indoor access points. As users spend the majority of their
online time in indoor areas, offloading mobile data traffic could
enable service providers to handle much more traffic. There are two
type of access point (AP) offloading indoor mobile data traffic,
called WiFi AP and femto AP, where the name `femto' comes from the
cell size. In general, the femto APs just cover small indoor areas
such as house or office. WiFi is the name of wireless LAN using the
IEEE 802.11 family of standard. WiFi APs have already been widely
deployed and exploited by users for data offloading due to the cheap
price and the capability of high data rate transmission.  However, in
view of QoS control and handoff, WiFi APs give limited support to
users. During moving from cellular networks to WiFi
connections or vice versa, which are called vertical handoff, significant delay is induced due to the
connecting to other types of networks. In addtion, turning on the
function of connecting to available WiFi APs in smart phones generate
more power consumption than turning off the function.  Thus, recently,
users who subscribe to unlimited data plan are willing to turn off
WiFi connection due to the poor QoS and power consumption.

On the other hand, femto APs exploit the same wireless access standard
with celluara networks. Moreover, the service providers can control both
femto APs and macro BSs. Therefore, handoff between macro and femto
cells or among femto cells can be supported by the service
provider. Besides, since the mobile devices need only one transmitter
to communicate both femto and macro cell, the power consumption is
also reduced compared to the case of accessing to WiFi APs. Therefore,
in view of data offloading, femto APs are better than WiFi APs,
because users do not hesitate to access to femtocell networks. Due to
the offloading, femto APs bring econmical benefit to both users and
service proivders\cite{SY11OC}. According to the femtocell
forum\cite{femto_forum}, the number of deployed femto access points
(APs) is expected to reach 49 million by 2014. Recent works also shows
that the economical benefit of femtocell networks.

In general, a small base station, called a femto AP, is deployed
manually in a home or office to build a femtocell, as shown in
Fig.~\ref{fig:interfemto}. Due to the fact that the deployment is not
organized by service providers but by users, deployment needs to be
easy such as \textit{`plug-and-play'} manners. Thus, femto APs are built so that they can configure
themselves\cite{CAG08FNS,HY10CL,JM09,HC09DOF}. In most service
scenarios, the femto AP connects to the core network by means of ISP
networks and sets its parameters, such as the channel and power level
for transmission, automatically.
However, although self-configuration makes femtocells user-friendly,
it has problems of its own. The main challenge that must be overcome
for self-configuration to work is interference caused to nearby users
that is induced by the transmission power of femto APs. When the power
level exceeds a certain threshold, nearby users who are not served by
the femto AP suffer from interference from the femto AP. On the other
hand, when the power level is lower than a certain threshold, the
femto AP cannot offer sufficient capacity to its serving users.  Thus,
developing an efficient power control scheme for femto APs requires
care. The issue of controlling the power of femto APs has been
considered in a number of previous works\cite{JM09,HC09DOF,CA09PC}. In
two of these works, distributed schemes are used to control the
transmission power to guarantee the service of the
macrocell\cite{JM09} or coverage over a designated area
 \cite{HC09DOF}. In the other paper \cite{CA09PC}, a game-theoretic approach
was used to construct a distributed power control scheme.

Previous works on controlling the transmission power of femto APs
focused on how to reduce interference or unwilling handoffs. However,
giving fair service to users is also an important
consideration. Femtocells with different numbers of users require more
throughput, as shown in Fig.~\ref{fig:interfemto}, where femto AP $i$
has more throughput than femto AP $j$ to give fair service. Herein, we
are dealing with the power control schemes which consider both
fairness and power saving. A
game-theoretic approach is used to formulate a distributed
power-control scheme that is both power-efficient and fair to
users. The main contributions of this paper are
summarized as follows:
\begin{compactenum}[\em 1)]
\item To consider both fairness and efficiency, we use
$\alpha$-fairness\cite{MW00FECC} to measure network performance. The
$\alpha$-fairness was defined with the $\alpha$-fair utility function,
which implicitly indicates the product of fairness and
efficiency\cite{KCS10ATF}.  
\item We desing a power control game. In our game model, femto APs control
transmission power without any massage passing between femto APs to
maximize their payoff function. The payoff function of each femto AP is based
on the number of users and the $\alpha$-fairness function where the range of $\alpha$ is from
0 to $\infty$.
\item We show both the uniqueness of the Nash
Equilibrium (NE) and the global stability of the NE when $\alpha \ge
1.$ Moreover, a counter example, which is unstalbe, is found where $\alpha=0.$  
\item By numerical results, we verify that the proposed power control
  scheme enhances fairness.
\end{compactenum}
While there exists a previous work \cite{ESD09DGT}, in the previous
work, proportional fairness function is considered only and the
scenarios where more than three femto APs are deployed to intefere
each other.

The rest of this chapter is organized as followings. Before
introducing the power control algorithm, we will define system measure
considering fairness in Section \ref{sec:system-measure}. In Section
\ref{sec:system-modelling}, a system model is described and
assumptions are introduced. A game model is suggested in Section
\ref{sec:game-theor-form}. By using the game model, a distributed
power control scheme is generated. The stability of the proposed power
control scheme is studied in Section \ref{sec:distr-power-contr}. The numerical results for the
scheme are given in Section \ref{sec:numerical-example}. In Section
\ref{sec:conclusions}, we summarize this chapter.

\begin{figure}[t]
  \centering
  \includegraphics[width=0.7\textwidth]{fig/femtointermodel}
  \caption{Femtocell networks and interference}
  \label{fig:interfemto}
\end{figure}

\section{System Measure}
\label{sec:system-measure}
Femtocells enhance system performance significantly. There are two
major reasons for that femtocells improve system utility. At first,
because wireless data traffic of macrocells is off-loaded to the
femtocells, the remaining users in macrocell can be served with the
higher throughput than before. In addition, since femtocells can
simultaneously transmit with macrocells, the total sum network
throughput of the system increases. However, how much of improvement
is achieved by means of femtocells is not clear. Thus, herein, we
will suggest how to measure the system performance.

In \cite{RFF09EE}, areal spectral efficiency is proposed for the
system performance metric, which is the mean of the achievable rates
in a network per unit bandwidth per unit area. However, the area
spectral efficiency is not suitable for our metric. Because the area
spectral efficiency only consider the system with the fixed number of
BSs, the efficiency cannot recognize the system enhancement due to the
increasing of access points from deployments of femto APs. For
example, when every spot has the same SINR and femto BSs are
deployed to have the SINR value in the femtocells, the area spectral
efficiency is not varied when the femto BSs are removed, whereas the
total system throughput is enhanced by frequency reuse. 

Some works, thus, have exploited total system efficiency for the
system metric. The total system efficiency can be defined as the sum
of the areal spectral efficiency for each cells. For example, if there
are 5 cells and the areal spectral efficiencies for the cells have the
same value $E$, the total system efficiency becomes $5 \times E.$
Because the total system efficiency increases as the number of cells
increases, it can measure the system enhancement due to reducing cell
size. However, still, the total system efficiency does not consider
areal fairness. When we compare two scenarios
that the first one is all of the area having the same throughput 5 and
the second scenario is the areal throughput being 10 or 0 with the
same chance, although both scenarios have the same total system
efficiency, the first one may be better than the second one.

In addition, we should consider load balancing. It is obvious that
the case that each cell has similar number of served users is better
than the case that some specific cells have dense and other cells are
sparse. In femtocell cases, in general, the macrocells have much more users
than femtocells have. Thus, the total system efficiency, in which the cell
throughput is just added without considering the number of served
users, may lose the load balancing effects. 

Therefore, herein, we
suggest metrics that consider followings:
\begin{compactenum}[\em 1)]
\item With the same BS deployment, the larger spectral efficiency
  means the better system.
\item When areal spectral efficiencies of two network
  topologies have the same distribution, the larger number of 
  BSs has the larger value.
\item Uniformly providing throughput for all spaces has the largest
  value among all cases having the same average throughput. 
\item If the distribution of users is proportional to the areal
  spectral efficiency of each cell, the system is in the most
  efficient case in view of load balancing.
\end{compactenum}

The proposed metric is a user-centric measure. Because of mobility of
users, they experience variety types of channel environments during
their move. Therefore, the expected value of the utility of the user
can show the network performance well. Moreover, in order to consider
fairness of area, the utility function uses $\alpha$-fairness
functions~\cite{MW00FECC}. The $\alpha$-fairness functions are
objective functions for optimization of which solution is fair and
efficient congestion controller. The metric $U$ for time duration $T$ is defined as
following:
\begin{eqnarray}
  U &=& \sum_i U_i , \\
  U_i &=& \int_T \frac{1}{1-\alpha}x_i^{1-\alpha}p_i(x_i,t)dt,
\end{eqnarray}
where $x_i$ is the throughput of user $i$ at the time $t$ and
$p_i(x_i,t)$ denotes the probability of user $i$ that the throughput
becomes $x_i$ at time $t.$ 
$p_i(x_i,t)$ is determined by the distribution of SINR, the number of
users in the serving cell, and the mobility pattern of the user. 


\section{System Modelling}
\label{sec:system-modelling}

Herein, we focus on scenarios in which femto APs are deployed in
private indoor areas, such as houses and apartments. Houses and
apartments are partitioned by walls or floor, so in general, users are
served by femto APs only when they are in their home. Thus, the number
of users that are served does not change often, if at all, and it is
unusual for a user to select a femto AP that is deployed in a
neighbor's home. In light of this, we assume that each femto AP knows
how many users it serves and that users are served by femto APs only
if they are in their home.

Furthermore, for the sake of simplicity, we make the following
assumptions to calculate the throughput of each user.  (1) We assume
that every user has sufficient data to transmit. (2) We also assume
that users experience the same level of interference when they are in
the same residence. (3) In a femtocell, each user gets the same
throughput during the service time. We hold these assumptions to be
realistic, for the following reasons: a) in practice, the demand for
data in wireless networks increases dramatically and quickly
approximates to the network capacity\cite{cisco_data}, b) over the
long term, users have almost identical channel environments in the
same residence, due to their mobility, and c) in view of long term
average, with fair scheduler, users have the same data rate.

Let $\mathcal{F}$ denote a set of femto APs and $N_i$ denote the
number of users served by the femto AP $i \in \mathcal{F}$. For
convenience of notation, the set of femto APs is represented as
an integer set $\{1,2,\cdots,|\mathcal{F}|\}$, where
$|\mathcal{F}|$ denotes the cardinality of $\mathcal{F}$. As shown in
Fig.~\ref{fig:interfemto}, the path loss between femto AP $i$ and
femto AP $j$ is denoted by $h_{ij}$ or $h_{ji}$, where the channel is
symmetric (i.e., $h_{ij}=h_{ji}$).  The notation used herein is
presented in Table~\ref{notations}.

The rate at which users who are served by femto AP $i \in
\mathcal{F}$ receive data can be expressed as follows :
%==============================================================
\begin{eqnarray}
R_{i} &=& \frac{1}{N_{i}}\log(1+\frac{h_{ii}p_i}{I + \sum_{j\neq i}
h_{ij}p_j}),
\end{eqnarray}
%==============================================================
where $I$ and $p_i$ represent noise power plus interference from the
macro BS and the transmission power of femto AP $i\in \mathcal{F}$,
respectively. Note that the transmission power of femto APs is
closely related to the performance of other femto APs.  

\subsection{$\alpha = 1$ (proportional fairness)}
Given that our goal is to increase fairness while using low
transmission power, we define a system performance measure as a
function of the transmission power vector $U({\bf p})$ based on
$\alpha-$fairness function. In the case of $\alpha=1$, the system
metric is represented as a proportional faireness function:
%==============================================================
\begin{eqnarray}
U({\bf p}) &=& \sum_{i \in \mathcal{F}} N_{i} \log (R_i)-\beta \sum_{i
  \in \mathcal{F}}  p_i, \label{eq:1}
\end{eqnarray}
%==============================================================
where {\bf p} denotes a power vector $[p_1 , p_2 , \cdots ,
p_{|\mathcal{F}|}]$. $\sum_{i \in \mathcal{F}} N_{i} \log (R_i)$
indicates the proportional fair function and power consumption is also
considered in the subtract term $\beta \sum_{i \in \mathcal{F}} p_i.$

\subsection{$\alpha$-fairness $( \alpha \neq 1)$}
Similarly, for the range $0 \le \alpha <1,$ the objective function can
be defined as a function of the transmission power vector $U({\bf p}).$
%==============================================================
\begin{eqnarray}
U({\bf p}) &=& \frac{1}{1-\alpha} \sum_{i \in \mathcal{F}} N_{i}  (R_i)^{1-\alpha}-\beta \sum_{i
  \in \mathcal{F}}  p_i, \label{eq:alpha01}
\end{eqnarray}
%==============================================================

\begin{table}[t!]
  \caption{Summary of Major Notation}
  \begin{centering}
    \begin{tabular}{|c|c|c|}
      \hline
      Notation & Description\tabularnewline
      \hline
      \hline
   $\mathcal{F}$ & set of femto APs \tabularnewline
\hline
 $N_i$ & number of users in $i$-th femto AP\tabularnewline
\hline
 $p_i$ & transmission power of $i$-th femto AP\tabularnewline
\hline
 $U_i(\cdot)$ & payoff function of $i$-th femto AP\tabularnewline
\hline
 $B_i(\cdot)$ & best response function of $i$-th femto AP\tabularnewline
\hline
$p_{-i}$ & power vector except for $i$ ($[p_1 , \cdots ,p_{i-1},p_{i+1},\cdots, p_{|\mathcal{F}|}]$)\tabularnewline
    \hline
${\bf p}$ & power vector ($[p_1 , p_2 , \cdots , p_{|\mathcal{F}|}]$)\tabularnewline
    \hline
${\bf B}({\bf p})$ & best response vector ($[B_{1} , B_{2} , \cdots , B_{|\mathcal{F}|}]$) for power vector ${\bf p}$ \tabularnewline
    \hline
\end{tabular}\label{notations}
    \par\end{centering}
\end{table}



\section{Game Theoretic Formulation}
\label{sec:game-theor-form}

We now present a game model that considers fairness
and the power consumption of femto APs. we also provide proofs
of the uniqueness and convergence of the NE.

\subsection{Game model}

A game model consists of {\bf players}, {\bf strategy sets} of
players, and {\bf payoff functions} for each player. In this work, we
model the power control of femto APs as a game. In the game model, the
femto APs control their transmission power so as to maximize their
utility function which is based on $\alpha-$fairness functions. Thus,
the {\bf players} and {\bf strategy sets} for this game are femto APs
and the range of transmission power, respectively. The remaining
element for this game is the payoff function.

% At first, we will study on the game under proportional fair utility
% function.
Our goal is to construct the payoff function so that the equilibrium
of the game converges to the maximum point of the $\alpha-$fairness
function.  However, finding the payoff function is not
simple. Moreover, since the femto APs are deployed in distributed
manner, it might be hard to exchange messages among femto APs. In
light of these considerations, we will suggest an heuristic game model
which does not need any information of other femtocells.  At first,
for the case of $\alpha=1,$ from Eq.~\eqref{eq:1}, we set the
\textbf{payoff function} $U_{i}$ for every player $i \in \mathcal{F}$
as follows:
%==============================================================
\begin{eqnarray}
U_{i}({\bf p})   &=&N_i \log(R_i)-\beta p_i\\
  &=& N_{i} \log(\log(1+\frac{h_{ii}p_i}{I + \sum_{j\neq i}
  h_{ij}p_j}))-N_{i}\log(N_{i})-\beta p_i. \label{eq:2}
\end{eqnarray}
%==============================================================
Note that our objective function is $U({\bf p}) = \sum_{i \in
  \mathcal{F}}U_{i}({\bf p})$. % Thus, when each femto AP tries to
% increase its payoff function, the object function might be also
% increased.
The payoff function is inspired from the objective function. We simply
decouple the objective function as sum of capacity equations of femto
APs, although the $R_i$ is varied by other femto APs'transmission
power. 

This utility function is simple to use and can be calculated without
any additional messages. In Eq.~\eqref{eq:2}, values of $N_i ,$ $h_{ii},$
and $p_{i}$ are known information for femto AP $i$, because the number
of users, channel information, and the transmission power of the femto
AP is determined by itself or easily get from its serving
users. Although the femto AP should exchange control messages with
other femto APs to know $h_{ij}$ and $p_{j}$, the denominator $I +
\sum_{j\neq i} h_{ij}P_j$ of Eq.~\eqref{eq:2} is just the noise
interference level of femto AP $i$. Therefore, by using this game
model, each femto AP can measure its payoff, which means each femto AP could update its
transmission power level to maximize its payoff value in a completely
distributed manner, under \textit{best response update} algoritm in
which players set their action so as to have the largest payoff.

The proposed game model may be summarized as follows:
\begin{separation}
{\it Game model : femto AP power game (proportional fair)}
\begin{compactenum}[\em i)]
    \item \textbf{player set :} the set of femto APs $\mathcal{F}$
    \item \textbf{strategy set of player $i$ :}  transmission power
      range $[p_{min} , p_{max}]$
    \item \textbf{payoff of player $i$ :} $N_{i} \log(\log(1+\frac{h_{ii}p_i}{I + \sum_{j\neq i}
  h_{ij}p_j}))-N_{i}\log(N_{i})-\beta p_i$
\end{compactenum}
\end{separation}
where $p_{min}$ and $p_{max}$ are the minimum and maximum available
power levels, respectively, for femto APs.

Similarly, for the fairness functions whose $\alpha$ range is not 1,
we can define a game model. From Eq.~\eqref{eq:alpha01}, we set the
\textbf{payoff function} $U_{i}$ for every player $i \in \mathcal{F}$
as follows:
%==============================================================
\begin{eqnarray}
U_{i}({\bf p})   &=&\frac{N_i (R_i )^{1-\alpha}}{1-\alpha}-\beta p_i\\
  &=&\frac{(N_{i})^{\alpha}}{1-\alpha} (\log(1+\frac{h_{ii}p_i}{I + \sum_{j\neq i}
  h_{ij}p_j}))^{1-\alpha}-\beta p_i. \label{eq:alphautility}
\end{eqnarray}
%==============================================================

The proposed game model for the fairness functions where the range of
$\alpha$ is not 1 may be summarized as follows:
\begin{separation}
{\it Game model : femto AP power game ($\alpha \neq 1$)}
\begin{compactenum}[\em i)]
    \item \textbf{player set :} the set of femto APs $\mathcal{F}$
    \item \textbf{strategy set of player $i$ :}  transmission power
      range $[p_{min} , p_{max}]$
    \item \textbf{payoff of player $i$ :} $\frac{(N_{i})^{\alpha}}{1-\alpha} (\log(1+\frac{h_{ii}p_i}{I + \sum_{j\neq i}
  h_{ij}p_j}))^{1-\alpha}-\beta p_i$
\end{compactenum}
\end{separation}

%========================================
% \begin{table}
% \caption{Game model}
%   \centering
%   \begin{tabular}{|c|c|}
%     \hline
%     \textbf{player set}& the set of femto APs $\mathcal{F}$  \tabularnewline
%     \hline
%     \textbf{strategy set of player $i$} & transmission power range $[p_{min} , p_{max}]$   \tabularnewline
% \hline
% \textbf{payoff of player $i$} & $N_{i} \log(\log(1+\frac{h_{ii}p_i}{I + \sum_{j\neq i}
%   h_{ij}p_j}))-N_{i}\log(N_{i})-\beta p_i$   \tabularnewline
% \hline
%   \end{tabular}

%   \label{tab:gamemodel}
% \end{table}
%========================================

\subsection{Supermodular Game} %\label{subsection1}

Due to the fact that supermodular games have some nice
characteristics, such as the convergence of the NE, such games have
been used in a number of previous works to develop power control
algorithms. To show how the work presented herein may be seen as
extending these earlier works, we show the supermodularity of the
proposed game. First, we verify that the proposed game is
supermodular. Then, by using the property of supermodular games such
that the NE is globally stable if the NE is unique, we prove the
convergence of the proposed distributed power control scheme. A
supermodular game is defined as follows.

\begin{separation}
{\it Definition 1 : Supermodular Game\cite{T98SC}}

Let $A_i$ and $\pi_i$ indicate a strategy set and payoff of player
$i$, respectively, where $\mathcal{N}:=\{1, ... , n\}$ denotes a set
of players. Let $A_{-i}$ denote a strategy vector set for every player
except player $i$.  The game is
supermodular if the following conditions are satisfied.
\begin{compactenum}[\em i)]
\item For each $i$, $A_i$ is a compact subset of $\mathbb{R}$ that
represents the set of all real numbers.nn
\item $\pi_i$ is upper semi-continuous in $a_i,a_{-i}$, where $a_i \in
  A_i$ and $a_{-i} \in A_{-i}$.
\item $\pi_i$ displays increasing differences in
($a_i,a_{-i}$) for all $a_i \in A_i$ and $a_{-i} \in A_{-i}$.  If $\pi_i$ is twice continuously
differentiable with $\frac{\partial^{2} \pi_i}{\partial
a_{i}\partial a_{j}}>0$ for all $i\neq j$, then $a_i$ displays
increasing differences in ($a_i,a_{-i}$).

\end{compactenum}
% \vskip1mm
\end{separation}


We now verify that the proposed game is supermodular. A game is
supermodular if and only if it possesses the following three
properties
\begin{compactenum}[\em i)]
\item $[p_{min}, p_{max}]$ is a compact subset of $\mathbb{R}$.

\item For all $p_{i} \in [p_{min}, p_{max}]$, the payoff function
  $U_{i}({\bf p})$
is continuous.

\item $\frac{\partial^{2} U({\bf p})}{\partial p_{i}\partial
p_{j}}>0$ for all $i,j \in \mathcal{F}$
\end{compactenum}
The proposed game has the above characteristics, which entails that
the game is supermodular. 
% ??(Actually, it doesn't. If you need to say "entail" or
% "imply", the above sentence
% should read "A game is
% supermodular if and only if
% it possesses the following
% three properties")??
The first and second properties can be identified easily. The third
property represents the increasing difference property. Let
$I_{t}=I+\sum_{j\neq i}h_{ji}P_{j}$, which gives the  total
interference arriving at $i \in \mathcal{F}$.  

\vspace{0.5cm}

\begin{lemma}\label{lem:incpro}
Payoff functions of femto APs are increasing difference when $\alpha = 1.$
\end{lemma}
\begin{proof}
A payoff function is increasing difference, when the partial differential equation of the payoff function of femto AP
$i$ with respect to $p_i$ and $p_j$ is positive. Thus, we will show
the positive. 

The partial
differential function of the payoff function can be expressed as
follows :
%=======================================
\begin{equation}
\frac{\partial U_{i}({\bf p})}{\partial p_i}=\frac{N_{i}h_{ii}}{(I_{t}
+h_{ii}p_{i})\log(1+\frac{h_{ii}p_i}{I_{t}})}-\beta
\end{equation}
%=======================================
Note that $I_{t}$ is easily measured by $i \in \mathcal{F}$ and the
payoff function for $i$ is determined by $I_{t}$ and $p_i$. Therefore,
the femto AP can find the maximum point of the payoff function without
any exchanging of messages with other femto APs.

For the sake of convenience of notation,  we set
$D=(I_{t}+h_{ii}p_{i})\log(1+\frac{h_{ii}p_i}{I_{t}})$. Then, the second
derivative of the payoff function is given by the following:
%========================================
\begin{equation}
\frac{\partial^{2} U_{i}({\bf p})}{\partial p_{i}\partial p_{j}}=\frac{-
N_{i}h_{ii}h_{ji}\{ \log(1+\frac{h_{ii}p_i}{I_{t}}) -
\frac{h_{ii}p_{i}}{I_{t}} \}}{D^{2}}.
\end{equation}
%========================================
Due to the fact that $\log(1+x)<x$ for all $x>0$
, we can confirm $\frac{\partial^{2}
U_{i}({\bf p})}{\partial p_{i}\partial p_{j}}>0$ because of $\frac{h_{ii}p_{i}}{A}>0$. Therefore, we can
conclude that \textbf{the proposed game is
a supermodular game}. 
\end{proof}

\vspace{0.5cm}
\begin{lemma}\label{lem:incal}
The proposed game has increasing difference property when $\alpha > 1.$
\end{lemma}
\begin{proof}
Similarly with the proof of Lemma~\ref{lem:incpro}, it is enough to show that the partial differential equation of the payoff function of femto AP
$i$ with respect to $p_i$ and $p_j$ is positive. 

The partial difference equation with respect to $p_j$  can be
expressed as follows :
%=======================================
\begin{equation} \label{eq:4}
\frac{\partial U_{i}({\bf p})}{\partial p_i}=\frac{(N_{i})^{\alpha}h_{ii}}{(I_{t}
+h_{ii}p_{i})}(\log(1+\frac{h_{ii}p_i}{I_{t}}))^{-\alpha}-\beta
\end{equation}
%=======================================
Then, Eq.~\eqref{eq:4} derivative with respective $p_i$ as the following:
%========================================
\begin{equation}
\frac{\partial^{2} U_{i}({\bf p})}{\partial p_{i}\partial p_{j}}=\frac{(N_{i})^{\alpha}h_{ii}h_{ji}(\log(1+\frac{h_{ii}p_i}{I_{t}}))^{-\alpha-1}}{(I_{t}
+h_{ii}p_{i})^2}\{\alpha \frac{h_{ii}p_i}{I_{t}} - \log(1+\frac{h_{ii}p_i}{I_{t}})  \}.
\end{equation}
%========================================
Due to the fact that $\log(1+x)<x$ for all $x>0$
, we can confirm $\frac{\partial^{2}
U_{i}({\bf p})}{\partial p_{i}\partial p_{j}}>0$ because of
$\frac{h_{ii}p_{i}}{A}>0$ when $\alpha >1$. 
\end{proof}

\vspace{0.5cm}

From Lemma~\ref{lem:incpro} and Lemma~\ref{lem:incal}, the proposed
game is supermodular game when $\alpha \ge 1.$ However, we cannot
guarantee that the game is supermodular when $\alpha <1.$ Due to the
fact, in the next section, the convergence of distributed power
control algorithm based on the proposed game is only guranteed when
$\alpha \ge 1.$

% This type of structure has several remarkable
%properties. The most noticeable property that can be applied to
%decentralized power control algorithm is that the unique NE of the
%supermodular game is globally stable.


\section{Distributed power control and stability}
\label{sec:distr-power-contr}
In this section, a distributed power control algorithm based on
proposed game will be suggested, and we will show that under what
condition the power control algorithm is \textit{stable} where
\textit{stable} means that the algorithm converge to unique stable
point from any initial state. 

\subsection{Best response update algorithm}

Every user updates his action to maximize his payoff, which is called
the best response update. The best response could have more than one
action, because there could be multiple maximum points in case of the
payoff function. In this case, users have multiple choices. On the
other hand, in a special case where the best response set has a unique
element for every case, users act deterministically when they act to
maximize their payoff. The first lemma asserts that the best response
sets for all users have a unique action. Thus, the best response can
be represented as a function of a power vector ${\bf p}$.
\vspace{0.5cm}
\begin{lemma}
The best response of the $i$-th femto AP, which is expressed as
\begin{eqnarray}B_{i}(p_{-i})&=& \arg \max_{p_i \in [P_{min},P_{max}]}
  U_{i}(p_i , p_{-i}),\end{eqnarray}
has a unique element for
every feasible $p_{-i}$. \label{lem:br}
\end{lemma}
\vspace{0.5cm}
The proof is in Appendix. Because the best response has only one
element, the best response update algorithm is
deterministic.

There are two types of best response based power control algorithm,
\textit{asynchronous} and \textit{synchronous}. Under
\textit{asynchronous} manner, each femto AP updates its transmission
power by its own timer. For example, with exponential clock, femto APs
control the power to maximize payoff only when their own clocks
expire. On the other hand, when by central controller or time
synchronization among femto APs, every femto APs can update the power
simultaneously, which is called \textit{synchronous}. 


\subsection{Convergence of NE when $\alpha \ge 1$}
We now show the uniqueness of the NE. In supermodular games,
uniqueness of the NE yields global stability, so a power control
scheme that is based on the proposed game guarantees convergence and
stability. Both \textit{asynchronous} and \textit{synchronous} are
proved the stability in this part.  All proofs are presented in the Appendix.

% Every user updates his action to maximize his payoff, which is called
% the best response update. The best response could have more than one
% action, because there could be multiple maximum points in case of the
% payoff function. In this case, users have multiple choices. On the
% other hand, in a special case where the best response set has a unique
% element for every case, users act deterministically when they act to
% maximize their payoff. The first lemma asserts that the best response
% sets for all users have a unique action. Thus, the best response can
% be represented as a function of a power vector ${\bf p}$.
% \begin{lemma}
% The best response of the $i$-th femto AP, which is expressed as
% \begin{eqnarray}B_{i}(p_{-i})&=& \arg \max_{p_i \in [P_{min},P_{max}]}
%   U_{i}(p_i , p_{-i}),\end{eqnarray}
% has a unique element for
% every feasible $p_{-i}$. \label{lem:br}
% \end{lemma}
% The proof is in Appendix.

Let ${\bf B}({\bf p})$ denote a best response vector for ${\bf{p}}$,
whose $i$-th element is $B_{i}(p_{-i})$. At each frame, the power
levels are changed according to ${\bf B}({\bf p})$. Lemma
\ref{lem:scal} states an important property called {\em Scalability},
which guarantees the convergence and stability of the
best-response-based update\cite{Y95FUPC}. {\em Scalability} is
strongly related to supermodularity.
\vspace{0.5cm}
\begin{lemma}
(Scalability) For all $\beta > 1$, $\beta {\bf B}({\bf p}) > {\bf B}(\beta {\bf p})$. \label{lem:scal}
\end{lemma}
\vspace{0.5cm}
The proof is in Appendix.

From the above lemmas, we conclude that the proposed game has a unique
NE that is globally stable. Uniqueness of the NE implies that
there is only one convergence point. Moreover, the
\textit{globally stable} property guarantees that the system converges
to the NE from any initial state. Thus, under the proposed game model,
the femtocells always operate in the unique NE.
\begin{theorem} \label{thm:uni} The proposed fair-power-control
  game has a unique NE and converges to the NE for any initial power
  vector, when $\alpha \ge 1$.
\end{theorem}
The proof is in Appendix.

\subsection{Divergence of $\alpha<1$}

Because the stability is guaranteed only for $\alpha \ge 1,$ stability
of $\alpha <1$ is not known yet. In this part, we will give an example
of $\alpha=0$
that is unstable.  Depend on the initial state of the femto networks,
the state of transmission power levels converges to different points. 

When $\alpha = 0,$ the utility functions of femto
APs become
\begin{eqnarray}
  U_{i}({\bf p})   &=&\log(1+\frac{h_{ii}p_i}{I + \sum_{j\neq i}
  h_{ij}p_j})-\beta p_i. \label{eq:3}
\end{eqnarray}
Then, the best response function which is the maximum point of
Eq.~(\ref{eq:3}) is represented as following : 
\begin{eqnarray}
B_{i}(p_{-i}) &=& \min \{ P_{max}, \max \{ P_{min}, \frac{1}{\beta} - \frac{I + \sum_{j\neq i}
  h_{ij}p_j}{h_{ii}} \} \} \\
 &:=& [ \frac{1}{\beta} - \frac{I + \sum_{j\neq i}
  h_{ij}p_j}{h_{ii}} ]_{P_{min}}^{P_{max}},
\end{eqnarray}
where $[\cdot]_{P_{min}}^{P_{max}}$ denotes that the upper bound is
$P_{max}$ and the lower bound is $P_{min}.$ Therefore, under the proposed scheme, femto APs update their
transmission power as water filling algorithm which fills power level
up to noise and interference values \cite{1262622}. The transmission
power increases as interference and noise level decreases and vice
versa. Let two femto APs be deployed and
$h_{12}=h_{21}=h_{11}=h_{22}=1.$ Under this topology, the power vector becomes $(
[\frac{1}{\beta} -I - p_2 ]_{P_{min}}^{P_{max}},[\frac{1}{\beta} -I -
p_1]_{P_{min}}^{P_{max}} ).$ Then, there are more than one
equilibriums such as $(P_{max},\frac{1}{\beta} -I - p_{max} )$ and
$(\frac{1}{\beta} -I - p_{max},P_{max} ),$ when $P_{min} \le
\frac{1}{\beta} -I - p_{max}.$ Therefore, we cannot guarantee the
stability of the proposed game based approach when $\alpha <1.$ 

% The proposed game may have multiple equilibrium. For example, let ass when
% $\alpha =0,$ according to the strating point of this game, the
% equilirium can be varied. 

\subsection{Power control and $\alpha$}

As varying $\alpha$, the power control algorithm has different
property. At first, as the $\alpha$ increases, the femto APs control
their transmission power so that SINR of each femto AP could have
similar value. As the $\alpha$ goes to infinity, the power control
algorithm becomes the algorithm that update power level so as to be a
target SINR. On the other hand, with the smaller $\alpha,$ the
capacities of femto APs becomes more unfair. The femto APs increase
their transmission power when the interference decreases. Thus, some
femtocells have high value of capacity but other femtocells suffer
from low quality of channel.

\section{Numerical Example}
\label{sec:numerical-example}

We have shown that the proposed game has a unique NE and is globally
stable. Thus, for every scenario, the proposed game guarantees a
deterministic result.  The performance of the proposed scheme is
compared with that of a conventional scheme.  \textit{In the case of
  the \textbf{conventional scheme}, femto APs set their transmission
  power level to support a target signal-to-interference noise ratio
  (SINR).} For example, when the target SINR of a femto AP is 10 and
the total interference arriving at the femto AP is 0.1, the femto AP
sets the transmission power level to 1. As we mentioned in previous
section, the conventional scheme is concurrent with the case that
$\alpha$ is infinity. In this numerical
analysis, 30dB and 40dB are used for the target SINRs\cite{YS10}.

In our simulations, the interference from neighbor macrocells is
ignored for the sake of simplicity. However, when neighbor macro BSs
use a different channel, which means that the reuse factor is not 1,
the intercell interference may be ignored. For a single macrocell
environment, we assume that the femtocells are distributed randomly
and uniformly in the macrocell area. For this environment, the
proportional fair values of femto users are measured by varying the
number of femtocells. The number of users of each femtocell is assumed
to be 1 or 2 with equal probability. For more detailed simulation
parameters and channel models, see \cite{2010arXiv1009.3522J}. From
the simulation, we get the following messages.

%=============================================
\begin{figure}[t!]
\begin{center}
\includegraphics[width=0.7\columnwidth]{fig/profair}
\caption{Proportional fair value vs number of femto cells }\label{fig:proportionalfair}
\end{center}
\end{figure}
%=============================================
%=============================================
\begin{figure}[t!]
\begin{center}
\includegraphics[width=0.7\columnwidth]{fig/tranpower}
\caption{Power consumption vs number of femto cells }\label{fig:betapower}
\end{center}
\end{figure}
%=============================================
%=============================================
\begin{figure}[t!]
\begin{center}
\includegraphics[width=0.7\columnwidth]{fig/capacity}
\caption{Total network capacity vs number of femto cells }\label{fig:targetpower}
\end{center}
\end{figure}
%=============================================

Fig.~\ref{fig:proportionalfair} and Fig.~\ref{fig:betapower} show the
proportional fair value and the average power consumption,
respectively, of the femto APs for varying numbers of femto cells. The
data presented in the figures show that \textbf{the proposed game
  model has slightly better proportional fair value and consumes much
  less power than the conventional model}. In Fig
~\ref{fig:proportionalfair}, when $\beta = 4$ and $\beta = 16$, the
proposed scheme outperforms the conventional scheme when the target
SINRs are 40dB and 30dB, respectively, while using much less
power. The main reason for the improvement in proportional fairness is that the proposed game
model is designed so that femtocells that serve a greater number of
users than average can use more power. The conventional scheme uses
more power because, under the conventional scheme, in the areas in
which the distribution of femto APs is dense, the femto APs increase
the amount of power they use to communicate with each other in order
to achieve the target SINR, whereas the proposed scheme restricts
power consumption in dense areas due to the utility function of the
proposed game model, in which the best response is a concave function
of the interference level.

Moreover, \textbf{the proposed scheme is stabler than the conventional
  scheme}. As the number of femtocells increases, the average power
consumption of the femto APs increases almost linearly in case of the
conventional scheme, whereas in the proposed scheme the average
transmission power remains almost constant. For instance, in the
case when the target SINR is 40dB, the average power consumption in
the conventional scheme changes from 0.44 to 0.75, whereas the
transmission power of the proposed scheme when $\beta$ is 4 changes
just from 0.43 to 0.5. Therefore, when a system operator wants to set
the average power consumption of femto APs to a target level, the
target can be met easily in the proposed scheme.
Fig.~\ref{fig:targetpower} shows that the proposed scheme performs
better than the conventional scheme with respect to total network
capacity (i.e. the sum of the capacity over all femtocells), although
the conventional scheme uses more transmission power than the
proposed scheme. 




\section{Conclusions}
\label{sec:conclusions}
Due to the fact that the deployment of femto APs is not organized by
service providers but by users, femtocell APs need to configure
themselves.  For example, the level of transmission power should be
determined without any exchange of messages among the APs.  Herein, we
have proposed a distributed power control method that consider
fairness. The scheme is based on a game model in which the payoff
function is designed as $\alpha$-fairness function, where the payoff
function is linearly increasing as the number of users
increases. In addition,
the payoff function has a transmission power term to restrict the
power consumption of the femto AP. We have shown that our game has the
following important properties, which guarantee the uniqueness and
stability of the NE.
\begin{compactenum}[\em i)]
\item The proposed game is supermodular, which implies that it has
NEs.
\item There is a unique NE and the NE is globally stable, which is
proved by {\em Scalability} and the supermodularity of the game.
\end{compactenum}

Moreover, through numerical examples, it has been shown that our
proposed power control scheme can be implemented in real environments
and can improve fairness among femtocells.

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